1. Field
The presently disclosed embodiment relates to the general field of the generation of radio signals under constraints. It relates more particularly to the field of adaptive radar emissions
2. Brief Description of Related Developments
With the advent of adaptive synthesis of transmit waveforms in radar and sonar, the problem of reconstructing signals with constant temporal envelope with prescribed Fourier transform has become of a prime importance.
In a known manner, a radar system is considered adaptive if one or more of its transmission and/or reception parameters are altered based on the operating environment.
The adaptation of transmit waveform has sparked considerable interest in recent times owing to the considerable performance benefit it can provide.
However, for maximum efficiency, the power amplifier of the radar is usually operated at saturation, which requires a temporal signal with a constant envelope. That's why the reconstruction of a constant envelope signal from the optimized transmission waveform of adaptive radar becomes an important subject of study.
From a mathematical point of view the formulation of the technical problem of the reconstruction of a transmission signal with a time constant envelope taking into account some given spectral constraints involves determining two functions Um(f) and ue(t) that satisfy the following equation:Um(f)ejθ(f)=F{ue(t)ejϕ(t)}  [1]
where F{.} denotes Fourier transform operation.
The function ue(t)ejϕ(t) describes the complex modulation of the transmission signal x(t), where ue(t) specifies the time-envelope of the waveform, which due to the radar system constraints must be a constant, say “A”.
The Fourier transform of ue(t)ejϕ(t) is denoted as Um(f)ejθ(f). Then, expressing relation [1] otherwise, ue(t)ejϕ(t) can be defined by the following relation:ue(t)ejϕ(t)=∫−∞∞Um(f)ej(θ(f)+2πft)df  [2]
Most of the adaptive transmit waveform solutions specify the modulus of the Fourier spectra, Um(f). The problem of finding θ(f) and/or ϕ(t), which meets both the desired time envelope constraint, ue(t) and the desired Fourier modulus spectra, Um(f) is called phase retrieval.
The question that arises here is to determine if it is possible to specify time envelope ue(t) and Fourier spectra Um(f) of a given signal independently, considering that the Fourier transform operation does seem to pose some constraint on the modulus of the Fourier pairs.
However, in a known manner, insofar as the time-bandwidth product of the signal becomes large, the bearing of each of the moduli of its Fourier pairs, ue(t) and Um(f), on the other one, tends to lose force.
To answer this question, several iterative known solutions have been proposed in the literature. One of the earliest is Gerchberg-Saxton algorithm (GSA), which is considered as a special case of Error Reduction Algorithm (ERA).
Based on ERA framework other general algorithms based on steepest-descent/conjugate-gradient, have also been proposed in literature.
Also, basic input-output (BIO) algorithm and Hybrid input-output (HIO) algorithms have been studied.
However, such alternating projection iterative algorithms usually suffer from slow convergence, convergence stagnation, permutation and scaling ambiguities, and sensitivity to initial seed (i.e. initial conditions).
U.S. Pat. No. 8,050,880 which is entitled “Generation of a constant envelope signal” formulates the problem of constant envelope radar signal with prescribed Fourier Transform Magnitude (FTM) in the alternating projection framework. It also outlines the fact that different starting points for the iterative algorithm gives rise to different solutions and their accuracy can be adjusted using a monotonic error criterion.